A generalized Cauchy dispersion formula and the refractivity of elemental semiconductors

A generalized Cauchy dispersion formula applicable to both normal and anomalous dispersion in the refractive index of transparent materials is derived from the Kramers-Kronig relations. Several commonly used dispersion formulae are closely related to this result. The expansion coefficients, heretofore considered empirical, are simply the odd moments of the absorption spectrum. These ideas are illustrated for natural diamond, and high-purity silicon and germanium. Analysis of published data for these materials discloses extrinsic dispersive effects in the far infrared, even for `high-purity' samples. We attribute this dispersion to free-carrier or defect-induced absorptions at energies below the range of measurements.